Download the ebook

The Poisson Distribution
Arnab Bhattacharyya∗
MIT Junior Lab
(Dated: 10/02/2003)
In this paper, we describe the theory of Poisson statistics. The issues and topics presented here are
widely applicable to many random processes in nature. In our experiment, we explore the statistics
of random events both by physical observations and by computer simulations. In particular, we
observe the statistical properties of γ-ray radiation incident on a scintillation counter. The results
obtained from this experiment are then compared to results from a Monte Carlo simulation of
Poisson processes.
1. INTRODUCTION n N . In general, without the approximations
in the last clause of the previous sentence, one
Random processes play a fundamental role has the simple binomial distribution:
in our physical world. Although the system N!
in consideration might be following determin- P (n) = pn (1 − p)N −n (1)
n!(N − n)!
istic classical mechanics, the large number of
When the conditions for the Poisson distribution
particles in our macroscopic universe leads to
are true, we can justiﬁably make the following
statistical variations in virtually every natural
approximations:
phenomenon. Random variables in nature usu-
ally have a probability distribution in the set of 1 N!
P (n) = pn (1 − p)N −n (2)
values they achieve. This probability distribu- n! (N − n)!
tion function varies in form for diﬀerent kinds 1 N!
≈ pn e−p(N −n) (3)
of systems. For example, the distribution of the n! (N − n)!
number of heads after a total number of N coin 1
≈ N n pn e−pN
tosses is peaked at N/2, whereas the probabil- n!
ity for surviving a game of Russian roulette de- λn e−λ
= (4)
creases exponentially with the number of rounds n!
played. In this paper, we examine a particular where we have substituted λ = N p, the mean
kind of probability distribution, the Poisson dis- value for n. Thus, we see that the probability
tribution, that arises very frequently in physical distribution is entirely independent of the proba-
measurements. bility for an individual occurrence. However, the
Poisson distribution is valid only for constant p
and for events that are uncorrelated with each
2. THEORETICAL DISCUSSION other.
The conditions described above for a Poisson
The Poisson distribution is a frequency dis- distribution arise in a wide range of ﬁelds. The
tribution for a discrete random variable. It de- Poisson distribution is often used to describe the
scribes the probability P (n) that an event char- number of occurences of events in a time inter-
acterized by a probability p independently oc- val. For a given time interval t, we can slice
curs n times in N trials, where p 1 and where it into small intervals of length ∆t. Then, it is
naturally true that the probability that an event
occurs in a ∆t interval is very small, while t/∆t
∗ Electronic address: [email protected] is very large. Each time interval ∆t can then be
2
regarded as independent trials and N = t/∆t is 3. EXPERIMENTAL OBSERVATIONS
the number of such trials. Then, we see that the
Poisson distribution applies. 3.1. γ-ray radiation counts
Other examples also abound. The distribu-
tions for the number of stars in a volume of In our experiment, the random process was
space, the number of cars that pass in front the incidence of γ-rays on a phosphorescent crys-
of a road sign, the number of mutations in a tal. As observed in the earlier section, such a
given stretch of DNA, and the number of soldiers random process follows a Poisson distribution.
killed by horse-kicks each year in the corps of Our basic experimental setup was as follows.
the Prussian cavalry (an example made famous A source of γ-ray radiation, in our case, sodium-
by an 1896 book of Ladislaus Bortkiewicz) are 22, was placed in front of a scintillator counter.
all roughly Poisson. In each of these cases, the The particles from this source impinged on a
events are mostly independent and the probabil- crystal of sodium iodide. The photons from the
ity of occurence is largely constant. Another im- phophorescence next hit the photocathode of
portant example is that the error in the counts the photomultiplier. The photoelectrically emit-
for a given bin in a histogram follows a Poisson ted electrons from the photocathode cascaded
distribution. As we shall see below, this means through a series of electrodes (called dynodes)
that we can ﬁnd the error bars for a histogram to create a measureable current. Finally, this
after having taken just one set of measurements current was ampliﬁed with a preampliﬁer and an
(in other words, just knowing the mean value for ampliﬁer to produce a pulse that could be mea-
the count at each bin). sured by a counter. So, our counts of pulses be-
The last statement can be justiﬁed by the fact ing output from the ampliﬁer reﬂected the pho-
that the standard deviation of data from a Pois- tons being radiated by the sodium source.
son process is dependent only on the mean of We measured the counts for mean rates of ap-
the data. More mathematically, we ﬁnd that proximately 1 sec−1 , 10 sec−1 , and 100 sec−1 .
∞ ∞ The histogram plots with superimposed Pois-
λn −λ λn
n = n e = λ e−λ = λ (5) son curves are shown on the next page. As can
n=0
n! n=0
n!
be seen from these plots, the histograms most
and also, more unexpectedly that, deﬁnitely are Poisson. Furthermore, the vari-
∞ ∞
ation in the shapes of the Poisson curves are
n
2 2λ −λ λn −λ very nicely illustrated here. When the mean is
n = n e = λ(n + 1) e
n=0
n! n=0
n! about 1, the distribution looks almost exponen-
= λ(λ + 1) (6) tial. When the mean increases to 100, the curve
is almost Gaussian.
Then, the variance is n2 − n 2 = λ, and so, the Although error bars are not shown on the his-
standard deviation of data in a random Poisson togram, as we mentioned above, the standard
process is simply the square root of the mean. error on each of the histogram bins is about the
As a ﬁnal comment in this section, we note square root of the height of the bin.
that as the mean λ becomes large, the Poisson
process more and more closely approximates a
Gaussian distribution. It’s important to note 3.2. Monte Carlo experiment
that this is not a special case of the central limit
theorem, which states that the distribution of To further verify the experimental results,
the mean approaches a Gaussian distribution as we also performed computer simulations of the
the sample size tends to inﬁnity. Poisson processes using the means above. The
3
simulation was a Monte Carlo simulation, where
we use the computer’s (pseudo)random num-
ber generator together with our knowledge of
the distribution function to create the speci-
ﬁed distribution of numbers. To be speciﬁc,
given a value of λ and a random number y, the
Monte Carlo simulator for the Poisson distri-
bution ﬁnds the smallest value of x for which
Pp (X) > y where
x
Pp (x) = p(x ) (7)
x =0
where p(x) is the Poisson probability distribu-
tion. The value of x thus found is the desired
simulated variate fo the Poisson process. Such a
simulation was done with the aid of a MATLAB
script, and the results can be compared to both
the theoretical model as well as the experimen-
tally determined values. The simulated curves
are also attached.
4. CONCLUSION
We conclude with the fact that the Poisson
curve is validated very well by our experiments,
both physical and computational. The experi-
ment with the radiation counts particularly well
illustrates the facts that the Poisson distribution
is found wherever we have independent events
with nonvarying probability of occurence.
FIG. 1: Experimentally determined distributions
with superimposed theoretical curves
4
FIG. 2: Monte Carlo simulations with superimposed
theoretical curves